length of the cardioid r = 1-cosx
- Thread starter kpx001
- Start date
You want arc length this time?. The formula for arc length in polar is
\(\displaystyle \int_{\alpha}^{\beta}\sqrt{r^{2}+\left(\frac{dr}{d{\theta}}\right)^{2}}d{\theta}\)
\(\displaystyle \int_{0}^{2\pi}\sqrt{(1-cos{\theta})^{2}+sin^{2}{\theta}}d{\theta}\)
But, \(\displaystyle (1-cos{\theta})^{2}+sin^{2}{\theta}=2-2cos{\theta}=2(1-cos{\theta})=4sin^{2}\frac{\theta}{2}\)
Make use of identities to simplify. Most times these things are set up so they simplify down to something easy to work with.
\(\displaystyle \int_{0}^{2\pi}\sqrt{4sin^{2}(\frac{\theta}{2})}d{\theta}=2\int_{0}^{2\pi}sin({\frac{\theta}{2})d{\theta}\)
\(\displaystyle \int_{\alpha}^{\beta}\sqrt{r^{2}+\left(\frac{dr}{d{\theta}}\right)^{2}}d{\theta}\)
\(\displaystyle \int_{0}^{2\pi}\sqrt{(1-cos{\theta})^{2}+sin^{2}{\theta}}d{\theta}\)
But, \(\displaystyle (1-cos{\theta})^{2}+sin^{2}{\theta}=2-2cos{\theta}=2(1-cos{\theta})=4sin^{2}\frac{\theta}{2}\)
Make use of identities to simplify. Most times these things are set up so they simplify down to something easy to work with.
\(\displaystyle \int_{0}^{2\pi}\sqrt{4sin^{2}(\frac{\theta}{2})}d{\theta}=2\int_{0}^{2\pi}sin({\frac{\theta}{2})d{\theta}\)
